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Contact three-manifolds
with exactly two simple Reeb orbits
Daniel Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings and Hui Liu |
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Geometry & Topology 27 (2023) 3801–3831
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Abstract |
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It is known that every contact form on a closed three-manifold has at least two simple Reeb orbits, and a generic contact form has infinitely many. We show that if there are exactly two simple Reeb orbits, then the contact form is nondegenerate. Combined with a previous result, this implies that the three-manifold is diffeomorphic to the three-sphere or a lens space, and the two simple Reeb orbits are the core circles of a genus-one Heegaard splitting. We also obtain further information about the Reeb dynamics and the contact structure. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation, the contact structure is universally tight, and in the case of the three-sphere the contact volume and the periods and rotation numbers of the simple Reeb orbits satisfy the same relations as for an irrational ellipsoid. |
Keywords
Reeb orbit, embedded contact homology, lens space,
pseudorotation
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Mathematical Subject Classification
Primary: 37J99, 53E50
Secondary: 53D42
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References |
Publication
Received: 29 September 2021
Revised: 24 January 2022
Accepted: 26 February 2022
Published: 5 December 2023
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Leonid Polterovich
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Authors |
| Daniel Cristofaro-Gardiner | ||
| Department of Mathematics University of California, Santa Cruz Santa Cruz, CA United States |
School of Mathematics Institute for Advanced Study Princeton, NJ United States |
Department of Mathematics University of Maryland College Park, MD United States |
| Umberto Hryniewicz | ||
| RWTH Aachen Aachen Germany |
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| Michael Hutchings | ||
| Department of Mathematics University of California, Berkeley Berkeley, CA United States |
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| Hui Liu | ||
| School of Mathematics and
Statistics Wuhan University Wuhan China |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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